It reflects a similar process in England where universities have been forced to draw up targets to boost the number of disadvantaged students admitted each year in return for retaining the power to charge up to £9,000 in tuition fees.

Prof Les Ebdon, head of the Government’s Office for Fair Access, has told the most sought-after universities to set the most “stretching” targets.

Speaking at a conference in London on Monday, he said the overall number of poor pupils admitted into higher education had increased by a third over the last decade but insisted numbers had failed to rise at the most prestigious institutions.

Data published earlier this year by the Higher Education Statistics Agency revealed that more than half of top universities recruited fewer pupils from the very poorest families in 2010/11. Two-thirds of institutions belonging to the elite Russell Group also recruited proportionally fewer state school students.

Prof Ebdon has previously suggested that universities can use “contextual data” – information on applicants’ family background and school performance – to make lower grade offers to disadvantaged applicants.

St Andrews, whose alumni include the Duke and Duchess of Cambridge, has faced fierce criticism north of the border for admitting only 14 children from the poorest backgrounds last year.

In a statement published on Tuesday, Mr Magee promised to redouble its efforts to accept more, but warned it faces a “considerable and continuing challenge” because so few school leavers in deprived areas are achieving even the most basic entry grades for a degree course.

“We have a choice – we can continue to beat up our leading universities for failing to admit more kids from our most deprived areas, or we can start, without shame or blame, to ask if perhaps there is something going wrong throughout the whole equation,” he said.

“We know that we could play the political game and change these figures overnight by lowering our entry grades, but experience tells us that we would simply be admitting these kids to fail, and that would be utterly dishonest.”

He said the problem could only be solved with a “concerted effort” to improve health, employment and housing as well as fostering a “culture of attainment” at all levels of education.

Mr Magee pointed to official figures showing only 220 children in the poorest communities north of the border last year achieved three As in their Higher exams, the Scottish equivalent of A-levels.

Of these, 55 applied to St Andrews and the university made offers to 34. However, only 14 accepted.

St Andrews also pointed out that competition for places is so fierce it has had to reject wealthier children who have achieved five good grades at school.

The SNP government in Scotland has threatened to introduce targets backed up by heavy fines if universities fail to make sufficient progress aimed at widening access

Under its deal, St Andrews has agreed next year to increase by 45 per cent the number of applicants admitted from the 20 per cent of poorest areas. However, this is the equivalent of only six more students.

Liz Smith, Scottish Tory education spokesman, said: “The message from St Andrews is clear… The real focus of the policy should be on raising standards and aspiration in every school.”

Addressing the Westminster Education Forum in London on Monday, Prof Ebdon said that universities in England were already “responding well” to OFFA’s reforms by “reaching out” to poor students.

But he insisted that the “challenge is with regard to the most selective universities”.

Prof Ebdon, former vice-chancellor of Bedfordshire University, said that students from the richest 20 per cent of households were seven times more likely to attend a top university than teenagers from the poorest 40 per cent.

“There are a multitude of reasons which contribute to this but it is clearly a serious challenge that we have to face,” he said.

OFFA has the power to fine institutions in England £500,000 or ban them from levying tuition fees of more than £6,000 a year – a potentially crippling sanction.

He said the watchdog had never used the powers and would seek to resolve stand-offs over admission targets “by negotiation”.

But he added: “I think the only circumstance… in which we would fine an institution is if they have made a promise to students which they did not fulfil. It does seem to me that part of the role of OFFA as regulator is to protect the interests of students.”

A Scottish Government spokesman said: “We have already taken action to ensure access to university is based on ability to learn, not the ability to pay by abolishing tuition fees which is delivering record numbers of students in higher education.

"We are now building on this by introducing a new system of student support that will ensure the most vulnerable students receive an income of at least £7,250 and all students can apply for increased student loans.

"We are also investing in the early years to improve the life chances of our children."